# Which method is efficient is compared to the other for numerical integration?

Out of the following methods for numerical integration which is one is best? I want to to know which of the following involves least amount of calculations. If someone can sort following in the order of increasing/decreasing efficiencies I would be much obliged. These methods are:

Newton-Cotes Integration formula.

The Trapizoidal Rule (Composite form).

Simpson's Rule (Composite form).

Romberg's Integration.

Double Integration.

Thank you!

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Dont use caps unless you want to shout –  Ram Feb 7 '13 at 12:15
Anyone else with his/her views? –  Taha Taha Feb 7 '13 at 12:34
Yes - the site moderators. Please do not use all caps, it is considered rude here. –  Zev Chonoles Feb 7 '13 at 13:45
Ok I will be careful next time. –  Taha Taha Feb 7 '13 at 15:10

The list doesn't make sense for ranking in terms of "efficiency". There are trade offs with some of them in terms of accuracy vs number of function evaluations.

First of all, "Double integration" doesn't even belong on this list, as it is not a numerical integration method.

The others I would rank in terms of just plain robustness, or what I would call, meaningful accuracy:

1) Romberg 2) trapezoidal rule 3) Simpson 4) Higher order Newton Cotes

By the way, Romberg is a Newton Cotes type of scheme, but simply involves applying the trapezoidal rule in a very smart way to almost eliminate error without oversampling. Many times, you do numerical integration using data obtained experimentally, so using a general Newton Cotes may represent fitting noise and thereby stymying accuracy gains from the higher orders. For simplicity, nothing beats plain trapezoidal. Simpson is also very good, but a little more complicated.

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Thanks a lot @rlgordonma –  Taha Taha Feb 7 '13 at 12:18
@TahaTaha: if you found the answer useful, please accept it so it will appear as such to the community. –  Ron Gordon Feb 7 '13 at 14:03
Brother, can you please elaborate it a little more, actually I have exam tomorrow morning and I have a question in my text "Compare the efficiency and characteristics of numerical integration methods" I found a lot online and could not find anything useful. Your answer has helped me quite but I think to be prepared fully I will need more arguments to answer my question carefully. If you can please do this I can mark your answer to be best. It is already quite good. Thanks! –  Taha Taha Feb 7 '13 at 14:19
If you want to assign a number to them: The $(m,n)$ Romberg is $O(1/2^{2 (m+1) n})$, trapezoidal rule is $O(1/(N^2))$, Simpson is $O(1/N^5)$ and Newton-Cotes can be pushed to whatever order you like, but keep in mind it is unstable: en.wikipedia.org/wiki/… –  Ron Gordon Feb 7 '13 at 14:27
Thank you very much! So nice of you. I have marked your question to be the best! –  Taha Taha Feb 7 '13 at 15:10